Question 33862
<pre><font size = 4><b>cos<font face = "symbol">q</font> = 5/12 and tan<font face = "symbol">q</font> is negative, find the exact value for csc<font face = "symbol">q</font>.

5/12 is a positive number and the cosine is positive in I and IV,
the tangent is negative in II and IV.  So, <font face = "symbol">q</font> can only be in IV.

Draw a picture          |
                        |
                        | x
                 -------|---------
                        |\   |
                        | \  |y
                        | r\ |
                        |   \|

Since cos<font face = "symbol">q</font> = 5/12 = x/r, put 5 for the x and 12 for the r

Draw a picture          |
                        |
                        | 5
                 -------|---------
                        |\   |
                        | \  |y
                        |12\ |
                        |   \|

Use the Pythagorean theorem 

                     x² + y² = r²
                     5² + y² = 12²
                     25 + y² = 144
                          y² = 119
                                 ___  
                           y = ±<font face = "symbol">Ö</font>119

Since y goes down, we select the negative sign.
                                 ___  
                           y = -<font face = "symbol">Ö</font>119

You are asked to find the csc<font face = "symbol">q</font>.
                   ___         ___
csc<font face = "symbol">q</font> = r/y = 12/(-<font face = "symbol">Ö</font>119) = -12/<font face = "symbol">Ö</font>119

Edwin
AnlytcPhil@aol.com</pre>